A characterization of ω-limit sets in subshifts of Baire space

2021 ◽  
Vol 500 (1) ◽  
pp. 125097
Author(s):  
Jonathan Meddaugh ◽  
Brian E. Raines
Keyword(s):  
Baire Space ◽  
Limit Sets

scholarly journals Characterization of topologically transitive attractors for analytic plane flows

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Martin Shoptrajanov ◽  
Nikita Shekutkovski
Keyword(s):  
Necessary Conditions ◽  
Topological Characterization ◽  
Plane Flow ◽  
Limit Sets ◽  
Topologically Transitive ◽  
Analytic Plane

We give necessary conditions for a set to be topologically transitive attractor of an analytic plane flow using topological characterization of ?-limit sets and the concept of upper semi-continuity of multi valued maps.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

scholarly journals A Characterization of theω-Limit Sets of Planar Continuous Dynamical Systems

1998 ◽  
Vol 145 (2) ◽  
pp. 469-488 ◽  
Author(s):  
Francisco Balibrea ◽  
Víctor Jiménez López
Keyword(s):  
Dynamical Systems ◽  
Limit Sets

scholarly journals Separate and Joint Properties of some Analogues of Pointwise Discontinuity

2014 ◽  
Vol 58 (1) ◽  
pp. 155-167
Author(s):  
Vasyl’ Nesterenko
Keyword(s):  
Metric Space ◽  
Baire Space ◽  
Residual Subset ◽  
Functions Of Two Variables ◽  
Joint Properties ◽  
Condition C

Abstract We study separate and joint properties of pointwise discontinuity, simple continuity and mild continuity of functions of two variables. In particular, it is shown that for a Baire space X, aBaire space Y which has a countable pseudobase and for a metric space Z, a function ƒ : X×Y → Z is pointwise discontinuous if and only if f satisfies (α, β)-condition and condition (C), and M = {x ∊ X : C(ƒx) = Y } is a residual subset of X. In addition, a characterization of simple continuity for mappings of one and two variables is given

Sequential discreteness and clopen-I-Boolean classes

1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty
Keyword(s):  
Final Section ◽  
Baire Space ◽  
Boolean Operations ◽  
Index Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Wadge Reducibility ◽  
Analytic Sets ◽  
Converse Results

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

ABSTRACT ω-LIMIT SETS

2018 ◽  
Vol 83 (2) ◽  
pp. 477-495 ◽  
Author(s):  
WILL BRIAN
Keyword(s):  
Dynamical System ◽  
Hausdorff Space ◽  
Complete Characterization ◽  
Topological Dynamics ◽  
Limit Set ◽  
Limit Sets ◽  
Shift Map ◽  
Image Position ◽  
Continuous Surjection

AbstractThe shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

A Characterization of ω-Limit Sets of Nonrecurrent Orbits in $\mathbb S^n$

2003 ◽  
Vol 13 (07) ◽  
pp. 1727-1732 ◽  
Author(s):  
Víctor Jiménez López ◽  
Gabriel Soler López
Keyword(s):  
Continuous Flow ◽  
Limit Set ◽  
Limit Sets ◽  
Continuous Flows ◽  
Smooth Flow

We characterize topologically ω-limit sets of nonrecurrent orbits for continuous flows on the n-sphere [Formula: see text]. Namely, it is shown that if Ω is the ω-limit set of some nonrecurrent orbit of a continuous flow on [Formula: see text] then it is the boundary of a region [Formula: see text] with connected complementary. Conversely, if Ω is the boundary of a region [Formula: see text] with connected complementary then there is a (C∞) smooth flow on [Formula: see text] having Ω as the ω-limit set of one of its nonrecurrent orbits.

Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows

2011 ◽  
Vol 54 (2) ◽  
pp. 311-315
Author(s):  
Habib Marzougui
Keyword(s):  
Topological Characterization ◽  
Surface Flows ◽  
Limit Sets ◽  
Continuous Flows ◽  
Orientable Surfaces

AbstractWe give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

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